   Chapter 19, Problem 65AP

Chapter
Section
Textbook Problem

Two coplanar and concentric circular loops of wire carry currents of I1 = 5.00 A and I2 = 3.00 A in opposite directions as in Figure P19.65. (a) If r1, = 12.0 cm and r2 = 9.00 cm, what are (a) the magnitude and (b) the direction of the net magnetic field at the center of the two loops? (c) Let r1, remain fixed at 12.0 cm and let r2 be a variable. Determine the value of r2 such that the net field at the center of the loop is zero. Figure P19.65

(a)

To determine
The magnitude of the net magnetic field.

Explanation

Given Info: The current in the outer circular loop is I1=5.00A . The current in the inner circular loop is I2=3.00A . The current in the two circular loops is in opposite direction. The radius of the outer circular loop is r1=12.0cm . The radius of the inner loop is r2=9.00cm .

Explanation:

The magnetic field at the center of a circular lop is given by,

B=μ0I2r       (1)

• I is the current in the loop
• r is the radius of the circular loop

Since the current in the loops are opposite in direction, considering the out of the page as positive direction, the net magnetic field will be given as,

Bnet=B2B1

• B2 is the field created by the inner loop
• B1 is the field created by the outer loop

Using (1) the net magnetic field will be,

Bnet=μ0I22r2μ0I12r1

Substitute 5.00A for I1 , 3.00A for I2 , 12.0cm for r1 , 9.00cm for r2 and 4π×107TmA-1 for μ0 to determine the net magnetic field,

Bnet=(4π×107TmA-1)(3

(b)

To determine
The direction of the net magnetic field.

(c)

To determine
The radius of the inner circular loop to have zero magnetic field at the center.

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